Hedging and value at risk: A semi‐parametric approach

The non‐normality of financial asset returns has important implications for hedging. In particular, in contrast with the unambiguous effect that minimum‐variance hedging has on the standard deviation, it can actually increase the negative skewness and kurtosis of hedge portfolio returns. Thus, the reduction in Value at Risk (VaR) and Conditional Value at Risk (CVaR) that minimum‐variance hedging generates can be significantly lower than the reduction in standard deviation. In this study, we provide a new, semi‐parametric method of estimating minimum‐VaR and minimum‐CVaR hedge ratios based on the Cornish‐Fisher expansion of the quantile of the hedged portfolio return distribution. Using spot and futures returns for the FTSE 100, FTSE 250, and FTSE Small Cap equity indices, the Euro/US Dollar exchange rate, and Brent crude oil, we find that the semiparametric approach is superior to the standard minimum‐variance approach, and to the nonparametric approach of Harris and Shen (2006). In particular, it provides a greater reduction in both negative skewness and excess kurtosis, and consequently generates hedge portfolios that in most cases have lower VaR and CVaR. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:780–794, 2010     

Minimum‐variance futures hedging under alternative return specifications

It is widely believed that the conventional futures hedge ratio, is variance‐minimizing when it is computed using percentage returns or log returns. It is shown that the conventional hedge ratio is variance‐minimizing when computed from returns measured in dollar terms but not from returns measured in percentage or log terms. Formulas for the minimum‐variance hedge ratio under percentage and log returns are derived. The difference between the conventional hedge ratio computed from percentage and log returns and the minimum‐variance hedge ratio is found to be relatively small when directly hedging, especially when using near‐maturity futures. However, the minimum‐variance hedge ratio can vary significantly from the conventional hedge ratio computed from percentage or log returns when used in cross‐hedging situations. Simulation analysis shows that the incorrect application of the conventional hedge ratio in crosshedging situations can substantially reduce hedging performance. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:537–552, 2005     

A note on estimating the minimum extended Gini hedge ratio

The extended Gini coefficient, Γ, is a measure of dispersion with strong theoretical merit for use in futures hedging. Yitzhaki (1982, 1983) provides conditions under which a two-parameter framework using the mean and Γ of portfolio returns yields an efficient set consistent with second-order stochastic dominance. Unlike mean-variance theory, the mean-Γ framework requires no particular return distribution or utility function to yield this conclusion. However, Γ must be computed iteratively making it less convenient to use than variance. Shalit (1995) offers a solution to the computation problem by suggesting an instrumental variables (IV) slope estimator, β^IV, as the basis for the minimum extended Gini hedge ratio where the instruments are based on the empirical distribution function (edf) of futures prices. However, the validity of employing the IV slope coefficient as the basis for the minimum extended Gini hedge ratio requires the questionable assumption that the rankings of futures prices to be the same as those for the profits of the hedged portfolio. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19:101–113, 1999     

A Markov regime switching approach for hedging stock indices

In this paper we describe a new approach for determining time‐varying minimum variance hedge ratio in stock index futures markets by using Markov Regime Switching (MRS) models. The rationale behind the use of these models stems from the fact that the dynamic relationship between spot and futures returns may be characterized by regime shifts, which, in turn, suggests that by allowing the hedge ratio to be dependent upon the “state of the market,” one may obtain more efficient hedge ratios and hence, superior hedging performance compared to other methods in the literature. The performance of the MRS hedge ratios is compared to that of alternative models such as GARCH, Error Correction and OLS in the FTSE 100 and S&P 500 markets. In and out‐of‐sample tests indicate that MRS hedge ratios outperform the other models in reducing portfolio risk in the FTSE 100 market. In the S&P 500 market the MRS model outperforms the other hedging strategies only within sample. Overall, the results indicate that by using MRS models market agents may be able to increase the performance of their hedges, measured in terms of variance reduction and increase in their utility. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:649–674, 2004     

An empirical analysis of dynamic multiscale hedging using wavelet decomposition

This study investigates the hedging effectiveness of a dynamic moving‐window OLS hedging model, formed using wavelet decomposed time‐series. The wavelet transform is applied to calculate the appropriate dynamic minimum‐variance hedge ratio for various hedging horizons for a number of assets. The effectiveness of the dynamic multiscale hedging strategy is then tested, both in‐ and out‐of‐sample, using standard variance reduction and expanded to include a downside risk metric, the scale‐dependent Value‐at‐Risk. Measured using variance reduction, the effectiveness converges to one at longer scales, while a measure of VaR reduction indicates a portion of residual risk remains at all scales. Analysis of the hedge portfolio distributions indicate that this unhedged tail risk is related to excess portfolio kurtosis found at all scales.     

Minimum variance cross hedging under mean‐reverting spreads,stochastic convenience yields,and jumps: Application to the airline industry

Exchange traded futures contracts often are not written on the specific asset that is a source of risk to a firm. The firm may attempt to manage this risk using futures contracts written on a related asset. This cross hedge exposes the firm to a new risk, the spread between the asset underlying the futures contract and the asset that the firm wants to hedge. Using the specific case of the airline industry as motivation, we derive the minimum variance cross hedge assuming a two‐factor diffusion model for the underlying asset and a stochastic, mean‐reverting spread. The result is a time‐varying hedge ratio that can be applied to any hedging horizon. We also consider the effect of jumps in the underlying asset. We use simulations and empirical tests of crude oil, jet fuel cross hedges to demonstrate the hedging effectiveness of the model. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:736–756, 2009     

Hedging long‐term commodity risk

This study focuses on the problem of hedging longer‐term commodity positions, which often arises when the maturity of actively traded futures contracts on this commodity is limited to a few months. In this case, using a rollover strategy results in a high residual risk, which is related to the uncertain futures basis. We use a one‐factor term structure model of futures convenience yields in order to construct a hedging strategy that minimizes both spot‐price risk and rollover risk by using futures of two different maturities. The model is tested using three commodity futures: crude oil, orange juice, and lumber. In the out‐of‐sample test, the residual variance of the 24‐month combined spot‐futures positions is reduced by, respectively, 77%, 47%, and 84% compared to the variance of a naïve hedging portfolio. Even after accounting for the higher trading volume necessary to maintain a two‐contract hedge portfolio, this risk reduction outweighs the extra trading costs for the investor with an average risk aversion. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:109–133, 2003     

Liability driven investment with alternative assets: Evidence from Brazil

We study the portfolio choice problem for an asset-liability investor who invests in stocks, equity mutual funds, government bonds, short term interest, hedge funds, listed real estate, and commodities futures available in Brazil. Inflation and real interest play as important risk sources. We estimate the asset classes and liabilities time-varying conditional covariance structure using an asymmetric multivariate dynamic conditional correlation GARCH model and compare the asset-liability portfolio's global minimum variance allocation with Brazilian pension funds' market portfolio. The conditional covariance structure provides insights about the complex dynamic relationships between the asset classes and liabilities. We find that some (though not all) Brazilian alternative assets render strong diversification and liabilities hedging benefits for asset-liability investors. There are significant strategic asset allocation differences between the market portfolio and the liability driven portfolio as given by our model. We, therefore, question the Brazilian pension funds' allocation.     

A copula‐based regime‐switching GARCH model for optimal futures hedging

The article develops a regime‐switching Gumbel–Clayton (RSGC) copula GARCH model for optimal futures hedging. There are three major contributions of RSGC. First, the dependence of spot and futures return series in RSGC is modeled using switching copula instead of assuming bivariate normality. Second, RSGC adopts an independent switching Generalized Autoregressive Conditional Heteroscedasticity (GARCH) process to avoid the path‐dependency problem. Third, based on the assumption of independent switching, a formula is derived for calculating the minimum variance hedge ratio. Empirical investigation in agricultural commodity markets reveals that RSGC provides good out‐of‐sample hedging effectiveness, illustrating importance of modeling regime shift and asymmetric dependence for futures hedging. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:946–972, 2009     

Use of high-frequency data to evaluate the performance of dynamic hedging strategies

The hedging performance results of generalized autoregressive conditional heteroskedasticity models are mixed; we address this herein by adopting an asymptotic setting to determine the relative performance of competing hedge ratios. The proxy variable is constructed through precise realized measures rather than through noisy squared returns because the substitution of the latent true hedged portfolio variance with a noisy proxy renders the loss function incapable of ranking forecasts consistently. The merits of allowing some features in modeling the spot–futures distribution are assessed. Empirical comparisons suggest that hedgers may favor the wrong model when the quality of the proxy variable deteriorates.     

Hedging and value at risk

In this article, it is shown that although minimum‐variance hedging unambiguously reduces the standard deviation of portfolio returns, it can increase both left skewness and kurtosis; consequently the effectiveness of hedging in terms of value at risk (VaR) and conditional value at risk (CVaR) is uncertain. The reduction in daily standard deviation is compared with the reduction in 1‐day 99% VaR and CVaR for 20 cross‐hedged currency portfolios with the use of historical simulation. On average, minimum‐variance hedging reduces both VaR and CVaR by about 80% of the reduction in standard deviation. Also investigated, as an alternative to minimum‐variance hedging, are minimum‐VaR and minimum‐CVaR hedging strategies that minimize the historical‐simulation VaR and CVaR of the hedge portfolio, respectively. The in‐sample results suggest that in terms of VaR and CVaR reduction, minimum‐VaR and minimum‐CVaR hedging can potentially yield small but consistent improvements over minimum‐variance hedging. The out‐of‐sample results are more mixed, although there is a small improvement for minimum‐VaR hedging for the majority of the currencies considered. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:369–390, 2006     

Estimation and hedging effectiveness of time‐varying hedge ratio: Flexible bivariate garch approaches

Bollerslev's ( 1990 , Review of Economics and Statistics, 52, 5–59) constant conditional correlation and Engle's (2002, Journal of Business & Economic Statistics, 20, 339–350) dynamic conditional correlation (DCC) bivariate generalized autoregressive conditional heteroskedasticity (BGARCH) models are usually used to estimate time‐varying hedge ratios. In this study, we extend the above model to more flexible ones to analyze the behavior of the optimal conditional hedge ratio based on two (BGARCH) models: (i) adopting more flexible bivariate density functions such as a bivariate skewed‐t density function; (ii) considering asymmetric individual conditional variance equations; and (iii) incorporating asymmetry in the conditional correlation equation for the DCC‐based model. Hedging performance in terms of variance reduction and also value at risk and expected shortfall of the hedged portfolio are also conducted. Using daily data of the spot and futures returns of corn and soybeans we find asymmetric and flexible density specifications help increase the goodness‐of‐fit of the estimated models, but do not guarantee higher hedging performance. We also find that there is an inverse relationship between the variance of hedge ratios and hedging effectiveness. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:71–99, 2010     

The Demand for Hedging with Futures and Options

The optimal hedging portfolio is shown to include both futures and options under a variety of circumstances when the marginal cost of hedging is nonzero. Futures and options are treated as substitute goods, and the properties of the resulting hedging demand system are explained. The overall optimal hedge ratio is shown to increase when the marginal cost of trading options is reduced. The overall optimal hedge ratio is shown to decrease when the marginal cost of trading futures is decreased. One implication is that hedging demand can be stimulated by a reduction in the perceived cost of trading options through the education of hedgers about options and the initiation of programs such as the Dairy Options Pilot Program. The demand approach is applied to estimate optimal hedge ratios for dairy producers hedging corn inputs in five regions of Pennsylvania. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:693–712, 2001     

Optimal hedge ratios in the presence of common jumps

This study derives optimal hedge ratios with infrequent extreme news events modeled as common jumps in foreign currency spot and futures rates. A dynamic hedging strategy based on a bivariate GARCH model augmented with a common jump component is proposed to manage currency risk. We find significant common jump components in the British pound spot and futures rates. The out‐of‐sample hedging exercises show that optimal hedge ratios which incorporate information from common jump dynamics substantially reduce daily and weekly portfolio risk. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:801–807, 2010     

Futures hedging using dynamic models of the variance/covariance structure

Dynamic futures‐hedging ratios are estimated across seven markets using generalized models of the variance/covariance structure. The hedging performances of the resultant dynamic strategies are then compared with static and naïve strategies, both in‐ and out‐of‐sample. Bayesian‐adjusted hedge ratios also are employed as error purgers. The empirical results indicate that the generalized dynamic models are well specified and that their use in determining optimal hedge ratios can lead to improvements in hedging performance as measured by the volatilities of the returns on the optimally hedged position. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:241–260, 2003     

Improved estimation of portfolio value‐at‐risk under copula models with mixed marginals

Portfolio value‐at‐risk (PVAR) is widely used in practice, but recent criticisms have focused on risks arising from biased PVAR estimates due to model specification errors and other problems. The PVAR estimation method proposed in this article combines generalized Pareto distribution tails with the empirical density function to model the marginal distributions for each asset in the portfolio, and a copula model is used to form a joint distribution from the fitted marginals. The copula–mixed distribution (CMX) approach converges in probability to the true marginal return distribution but is based on weaker assumptions that may be appropriate for the returns data found in practice. CMX is used to estimate the joint distribution of log returns for the Taiwan Stock Exchange (TSE) index and the associated futures contracts on SGX and TAIFEX. The PVAR estimates for various hedge portfolios are computed from the fitted CMX model, and backtesting diagnostics indicate that CMX outperforms the alternative PVAR estimators. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:997–1018, 2006     

Intraday return predictability,portfolio maximisation,and hedging

We examine whether intraday Chinese return predictability is linked to optimal portfolio holding and hedging. We find that: (1) S&P500 futures returns only predict Chinese spot market returns in up to 5-minute of trading with predictability disappearing at higher frequencies of trade; (2) the portfolio weight is maximised at the 5-minute trading frequency, when predictability is the strongest; and (3) when predictability is the strongest, significantly less shorting of the futures is required to minimise risk when a long position is taken in the Chinese market.     

Dynamic hedging with futures: A copula‐based GARCH model

In a number of earlier studies it has been demonstrated that the traditional regression‐based static approach is inappropriate for hedging with futures, with the result that a variety of alternative dynamic hedging strategies have emerged. In this study the authors propose a class of new copula‐based GARCH models for the estimation of the optimal hedge ratio and compare their effectiveness with that of other hedging models, including the conventional static, the constant conditional correlation (CCC) GARCH, and the dynamic conditional correlation (DCC) GARCH models. With regard to the reduction of variance in the returns of hedged portfolios, the empirical results show that in both the in‐sample and out‐of‐sample tests, with full flexibility in the distribution specifications, the copula‐based GARCH models perform more effectively than other dynamic hedging models. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1095–1116, 2008     

Economic uncertainty,trading activity,and commodity futures volatility

This paper investigates the dynamics of commodity futures volatility. I derive the variance decomposition for the futures basis and show unexpected excess returns result from new information about expected future interest rates, convenience yields, and risk premia. Measures of uncertainty in economic conditions have significant predictive power for realized volatility of commodity futures returns, after controlling for lagged volatility, returns, commodity index trading, hedging pressure, and other trading activity, even during the so-called “index financialization” period. During this period, hedge fund performance predicts volatility in grain commodities, which are affected by the US ethanol mandate.     

Fractional cointegration and futures hedging

This article examines the performance of various hedge ratios estimated from different econometric models: The FIEC model is introduced as a new model for estimating the hedge ratio. Utilized in this study are NSA futures data, along with the ARFIMA-GARCH approach, the EC model, and the VAR model. Our analysis identifies the prevalence of a fractional cointegration relationship. The effects of incorporating such a relationship into futures hedging are investigated, as is the relative performance of various models with respect to different hedge horizons. Findings include: (i) Incorporation of conditional heteroskedasticity improves hedging performance; (ii) the hedge ratio of the EC model is consistently larger than that of the FIEC model, with the EC providing better post-sample hedging performance in the return–risk context; (iii) the EC hedging strategy (for longer hedge horizons of ten days or more) incorporating conditional heteroskedasticty is the dominant strategy; (iv) incorporating the fractional cointegration relationship does not improve the hedging performance over the EC model; (v) the conventional regression method provides the worst hedging outcomes for hedge horizons of five days or more. Whether these results (based on the NSA index) can be generalized to other cases is proposed as a topic for further research. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 457–474, 1999